Is it permissible to pray sitting down and to wipe over the feet in an unclean workplace, and what is the reason for the lukewarmness in prayer at home, and does my wife have the right to urge me to pray?

Your work must be permissible, with extreme caution regarding illicit gains. It is not lawful for a Muslim to perform tayammum (dry ablution) when water is available, nor to wipe over the feet when capable of washing them. You can perform wudu' (ablution) before work or leave work to perform wudu' and prayer. Wiping over socks is permissible for one who performed a complete wudu' and then put them on while in a state of purity, or for one who gave their feet a light wash. It is not permissible to perform an obligatory prayer while sitting if one is able to stand, as standing is a pillar of prayer, and your prayer is invalid if you pray sitting while able to stand. Purity, removal of ritual impurity, and removal of physical impurity are two conditions for prayer. You must pay attention to prayer and be keen to perform it in the mosque due to the great merit therein. What your wife does by urging you to establish prayer is her duty, and it is evidence of the strength of her faith and the soundness of her intellect.